Homology of Smooth Splines: Generic Triangulations and a Conjecture of Strang

Abstract

ABSTRACT. For A a triangulated d-dimensional region in Kd, let S^(A) denote the vector space of all Cr functions F on A that, restricted to any simplex in A, are given by polynomials of degree at most rn. We consider the problem of computing the dimension of such spaces. We develop a homological approach to this problem and apply it specifically to the case of triangulated manifolds A in the plane, getting lower bounds on the dimension of S^(A) for all r. For r = 1, we prove a conjecture of Strang concerning the generic dimension of the space of C1 splines over a triangulated manifold in R2. Finally, we consider the space of continuous piecewise linear functions over nonsimplicial decompositions of a plane region.

A C1-P2 finite element without nodal basis

“…A vertex v is called singular if v is an interior vertex, if exactly four edges meet at v, and if the four edges form two straight lines. The Strang's conjecture is the equality in (3.8), under certain conditions (see [2,13,14] and references cited there). Nevertheless, the equality for (3.8) has been proved in [14] for C 1 -P 2 elements on the criss-cross grid ( Fig.…”

“…These (n + 2) 2 basis functions {φ jk } are linearly independent (on its supported domain (−h, 1 + h) 2 . But when restricted in the sub-domain (0, 1) 2 , they are not, because (see (3.2))…”

A Homological Interpretation of Skeletal Rigidity

“…We thank Henry Crapo [6] and Lou Billera [2] for demonstrating in convincing ways that the homology of an appropriate complex is the essential tool for clarifying the geometry and combinatorics of such an geometric problem. We also thank Carl Lee and Lou Billera for ongoing conversations on the combinatorics and geometry of these patterns.…”

An algorithm for constructing a basis for 𝐶^{𝑟}-spline modules over polynomial rings

“…With pointwise operations of addition and multiplication the set C r (2) forms a ring and the polynomial ring R is just a subring of C r (2). Hence C r (2) is an R-module in a natural way.…”